Definition:Ordered Pair

Definition

The definition of a set does not take any account of the order in which the elements are listed.

That is, $\left\{{a, b}\right\} = \left\{{b, a}\right\}$, and the elements $a$ and $b$ have the same status - neither is distinguished above the other as being more "important".

An ordered pair is a two-element set together with an ordering.

In other words, one of the elements is distinguished above the other - it comes first.

Such a structure is written:

$\left({a, b}\right)$

and it means:

first $a$, then $b$.

Some sources call this just a pair, taking the fact that it is ordered for granted.

Kuratowski Formalization

The concept of an ordered pair can be formalized by the definition:

$\left({a, b}\right) = \left\{{\left\{{a}\right\}, \left\{{a, b}\right\}}\right\}$

This formalization justifies the existence of ordered pairs in Zermelo-Fraenkel set theory.

The elements of an ordered pair are called its coordinates.

Alternative Notation

In the field of symbolic logic and modern treatments of set theory, the notation $<a, b>$ is often seen to denote an ordered pair.

This notation is found in many textbooks and journal articles in set theory, including the widely referenced textbooks of Herbert B. Enderton and Patrick Suppes.

Some users even claim that $<a \ b>$ is the way to go, but such seem still to be in a minority.

Also see

• Cartesian Product

• Equality of Ordered Pairs

• Ordered Tuple as Ordered Set

Sources

• Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts (1951): Introduction $\S 2$
• Paul R. Halmos: Naive Set Theory (1960)... (previous)... (next): $\S 6$: Ordered Pairs
• W.E. Deskins: Abstract Algebra (1964): $\S 1.2$
• J.A. Green: Sets and Groups (1965)... (previous)... (next): $\S 1.7$
• Seth Warner: Modern Algebra (1965): $\S 1$
• Richard A. Dean: Elements of Abstract Algebra (1966): $\S 0.2$
• George McCarty: Topology: An Introduction with Application to Topological Groups (1967): Introduction
• Ian D. Macdonald: The Theory of Groups (1968): Appendix
• B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra (1970): $\S 1.1$
• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 9$
• T.S. Blyth: Set Theory and Abstract Algebra (1975): $\S 3$
• Gary Chartrand: Introductory Graph Theory (1977): Appendix $\text{A}.2$
• Thomas A. Whitelaw: An Introduction to Abstract Algebra (1978)... (previous)... (next): $\S 8$
• Keith Devlin: The Joy of Sets: Fundamentals of Contemporary Set Theory (1993): $\S 1.4$